Space of modular forms of level 1664, weight 2, and trivial character

• Label: 1664.2.a
• Level: $1664$
• Weight: $2$
• Character orbit: 1664.a
• Rep. character: $\chi_{1664}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $28$
• Sturm bound: $448$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(448\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1664))\).

	Total	New	Old
Modular forms	240	48	192
Cusp forms	209	48	161
Eisenstein series	31	0	31

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(54\)	\(10\)	\(44\)		\(47\)	\(10\)	\(37\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)		\(66\)	\(16\)	\(50\)		\(58\)	\(16\)	\(42\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(66\)	\(14\)	\(52\)		\(58\)	\(14\)	\(44\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(54\)	\(8\)	\(46\)		\(46\)	\(8\)	\(38\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(108\)	\(18\)	\(90\)		\(93\)	\(18\)	\(75\)		\(15\)	\(0\)	\(15\)
Minus space		\(-\)		\(132\)	\(30\)	\(102\)		\(116\)	\(30\)	\(86\)		\(16\)	\(0\)	\(16\)

Trace form

\( 48 q + 48 q^{9} + 48 q^{25} + 32 q^{33} + 32 q^{41} + 48 q^{49} + 32 q^{57} + 144 q^{81} + 64 q^{89} + 64 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1664))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
1664.2.a.a	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.a	$1$	$2$	\(0\)	\(-3\)	\(-3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-3q^{5}-q^{7}+6q^{9}-4q^{11}+\cdots\)
1664.2.a.b	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
1664.2.a.c	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+2q^{7}+q^{9}-q^{13}+\cdots\)
1664.2.a.d	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+4q^{7}+q^{9}-6q^{11}+\cdots\)
1664.2.a.e	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}-4q^{7}+q^{9}-6q^{11}+\cdots\)
1664.2.a.f	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}-2q^{7}+q^{9}+q^{13}+\cdots\)
1664.2.a.g	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}-2q^{9}+q^{13}+q^{15}+\cdots\)
1664.2.a.h	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.h	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}-2q^{9}+4q^{11}+q^{13}+\cdots\)
1664.2.a.i	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}-q^{13}-q^{15}+\cdots\)
1664.2.a.j	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}+4q^{11}-q^{13}+\cdots\)
1664.2.a.k	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.h	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}-2q^{9}-4q^{11}+q^{13}+\cdots\)
1664.2.a.l	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.g	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}-2q^{9}+q^{13}-q^{15}+\cdots\)
1664.2.a.m	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.h	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}-2q^{9}-4q^{11}-q^{13}+\cdots\)
1664.2.a.n	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.g	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}-2q^{9}-q^{13}+q^{15}+\cdots\)
1664.2.a.o	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.d	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}-4q^{7}+q^{9}+6q^{11}+\cdots\)
1664.2.a.p	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}-2q^{7}+q^{9}-q^{13}+\cdots\)
1664.2.a.q	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+2q^{7}+q^{9}+q^{13}+\cdots\)
1664.2.a.r	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.d	$1$	$0$	\(0\)	\(2\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
1664.2.a.s	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{5}+q^{7}+6q^{9}+4q^{11}+\cdots\)
1664.2.a.t	$1664$	$2$	1664.a	1.a	$1$	$1$	$1$	$13.287$	\(\Q\)	None	✓	✓			1664.2.a.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
1664.2.a.u	$1664$	$2$	1664.a	1.a	$1$	$2$	$2$	$13.287$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		1664.2.a.u	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
1664.2.a.v	$1664$	$2$	1664.a	1.a	$1$	$2$	$2$	$13.287$	\(\Q(\sqrt{2}) \)	None	✓	✓			1664.2.a.u	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
1664.2.a.w	$1664$	$2$	1664.a	1.a	$1$	$2$	$2$	$13.287$	\(\Q(\sqrt{2}) \)	None	✓	✓			1664.2.a.u	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
1664.2.a.x	$1664$	$2$	1664.a	1.a	$1$	$2$	$2$	$13.287$	\(\Q(\sqrt{2}) \)	None	✓	✓			1664.2.a.u	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
1664.2.a.y	$1664$	$2$	1664.a	1.a	$1$	$5$	$5$	$13.287$	5.5.592456.1	None	✓	✓	✓		1664.2.a.y	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-5\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{3}q^{5}+(-1-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
1664.2.a.z	$1664$	$2$	1664.a	1.a	$1$	$5$	$5$	$13.287$	5.5.592456.1	None	✓	✓			1664.2.a.y	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(5\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{3}q^{5}+(1+\beta _{2}-\beta _{3}-\beta _{4})q^{7}+\cdots\)
1664.2.a.ba	$1664$	$2$	1664.a	1.a	$1$	$5$	$5$	$13.287$	5.5.592456.1	None	✓	✓	✓		1664.2.a.y	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(5\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{3}q^{5}+(1+\beta _{2}-\beta _{3}-\beta _{4})q^{7}+\cdots\)
1664.2.a.bb	$1664$	$2$	1664.a	1.a	$1$	$5$	$5$	$13.287$	5.5.592456.1	None	✓	✓			1664.2.a.y	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-5\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{3}q^{5}+(-1-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1664))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1664)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(832))\)\(^{\oplus 2}\)